Worked example 1
What volume does 2.50 mol of nitrogen occupy at 1.20 atm and 35 °C?
Try it first: Convert the temperature and rearrange the equation before you reach for a calculator.
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Gases, Solutions & SpectroscopyIdeal Gas LawContent level: Core 20 min
The Ideal Gas Law
What you'll be able to do: Use PV = nRT with consistent units to find any one gas variable, and derive the simple gas laws from it.
Introduction
One equation replaces every named gas law you may have memorised. Get the units right and PV = nRT will answer almost any single-state gas question.
These are recommended, not required. You can start this lesson at any time.
Learning objectives
- Apply PV = nRT with a consistent set of units
- Choose the correct value of R for the pressure unit given
- Derive the Boyle, Charles and Avogadro relationships from the ideal gas law
- Use the two-state form to handle a change in conditions
Lesson
The equation and its constants
PV = nRT relates the four state variables of a gas sample. Use R = 0.08206 L*atm/(mol*K) when pressure is in atmospheres, or R = 8.314 J/(mol*K) when you need energy units. Temperature is always in kelvin, obtained by adding 273.15 to a Celsius value.
PV = nRT
Mixing kPa with the 0.08206 value of R is the single most common error in this unit. Check the units of R against the units of P before substituting.
Recovering the simple gas laws
Hold n and T constant and PV is a constant, which is the law of Boyle. Hold n and P constant and V/T is a constant, which is the law of Charles. Hold P and T constant and V/n is a constant, which is the law of Avogadro. You never need to memorise them separately.
Changing conditions
When the same sample moves between two states, divide one form of the equation by the other. Anything held constant cancels, leaving P₁V₁/T₁ = P₂V₂/T₂. If moles also change, keep n in the expression as P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂).
P₁V₁/T₁ = P₂V₂/T₂
Molar mass and density
Substituting n = m/M into PV = nRT gives PM = dRT, where d is density in grams per litre. This is how a gas density measurement identifies an unknown gas, and it explains why warm air, at lower density, rises.
Audit a gas-law calculation with Kelvin
When the amount and volume of a gas stay fixed, P/T is constant, with temperature measured in kelvin. A Celsius ratio gives a false result because 0 °C is not zero thermal motion. State what is held constant and convert temperatures before using a proportional shortcut; otherwise return to PV = nRT.
Only absolute temperature can appear in a gas-law ratio.
Key ideas
Definition
Standard conditions
At STP, defined as 273.15 K and 1 atm, one mole of ideal gas occupies 22.4 L.
Rule
Kelvin only
Every gas law calculation uses absolute temperature; Celsius values give nonsense ratios.
Key concept
State equation
PV = nRT describes one state of a sample, so a change needs the equation applied twice.
Equation
Density form
PM = dRT lets a density measurement give a molar mass.
Equation
Ideal gas law
PV = nRT
- P = pressure, atm
- V = volume, L
- n = moles
- R = 0.08206 L*atm/(mol*K)
- T = kelvin
Equation
Density form
PM = dRT
- M = molar mass, g/mol
- d = density, g/L
Worked examples
Worked example 2
A gas has a density of 1.96 g/L at 1.00 atm and 273 K. Identify its molar mass.
Try it first: Choose between PV = nRT and PM = dRT based on the data you were given.
0 of 4 steps revealed.
Worked example 3
A rigid tank is at 2.00 atm and 300 K. What is its pressure at 450 K if no gas escapes?
Try it first: The volume and amount are unchanged.
0 of 3 steps revealed.
Common mistakes
Leaving temperature in °C.
Why it's wrong: Ratios of Celsius values are not ratios of absolute temperature, so the answer can even be negative.
Check instead: Add 273.15 as the very first step of every gas problem.
Using 22.4 L/mol at conditions other than STP.
Why it's wrong: The molar volume changes with both pressure and temperature.
Check instead: Use PV = nRT unless the problem states standard conditions.
Pairing R = 0.08206 with pressure in kPa or mmHg.
Why it's wrong: The units of R must match the units substituted, or the result is off by a large factor.
Check instead: Convert pressure to atm, or switch to a value of R with matching units.
After the lesson · Question bank
Independent practice
Work through new problems on this topic on your own. The lesson's quick checks and worked examples stay above if you need to review a step.
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What you should now know
The ideal gas law links pressure, volume, moles and temperature through the gas constant R. With R = 0.08206 L*atm/(mol*K), pressure must be in atmospheres, volume in litres and temperature in kelvin. Holding two variables fixed recovers the laws of Boyle, Charles and Avogadro, and the two-state form P₁V₁/T₁ = P₂V₂/T₂ handles changes.
- PV = nRT covers every single-state gas calculation
- Match the value of R to your pressure unit
- Temperature is always in kelvin
- P₁V₁/T₁ = P₂V₂/T₂ handles changes in conditions
- PM = dRT converts a gas density into a molar mass
Sources and further reading
This lesson is original Chem Help content. No external sources were adapted.